Proc IMechE Part G: Elastoplastic buckling analysis of ...profdoc.um.ac.ir/articles/a/1047922.pdf · Elastoplastic buckling analysis of rectangular thick plates by incremental and

Original Article

Elastoplastic buckling analysis ofrectangular thick plates by incrementaland deformation theories of plasticity

M Maarefdoust and M Kadkhodayan

Abstract

In this study, the elastoplastic buckling behavior of thick plates under uniaxial, biaxial compression/tension, pure shear

and combined shear, and uniaxial compression loading are analyzed using a stability analysis based on generalized

differential quadrature method. Based on first-order shear deformation plate theory and incremental and deformation

theories of plasticity, the nonlinear equilibrium equations are developed. Various combinations of clamped, simply sup-

ported, and free boundary conditions are considered. The governing equations and solution domain are discretized by

the generalized differential quadrature method. The results are compared with known solutions in literature to verify the

established methodology and procedures. The effects of aspect, loading and thickness ratios, shear correction factor,

different boundary conditions, and behavior of material on the buckling coefficient are discussed. Finally, some mode

shapes of various loading and boundary conditions are illustrated.

Keywords

Pure shear loading, generalized differential quadrature method, combined loading, elastoplastic buckling, first-order shear

deformation theory

Date received: 11 November 2013; accepted: 12 August 2014

Introduction

Elastic and plastic buckling of plate have been widelyinvestigated by researchers numerically and analytic-ally in last decades. The plastic buckling of plates havebeen discussed and studied using both incrementaltheory (IT) and deformation theory (DT) of plasticity.In thick plates, the shear deformation theories ofplates such as Reissner–Mindlin theory has to beused to consider the effect of transverse shear deform-ation. Nevertheless, most of studies have been basedon the classical plate theory (CPT) which neglects theeffect of transverse shear deformation and lead tosome inconsistencies in the results when the thicknessof plate increases.

Handelman and Prager1 developed the fundamen-tal equations for the plastic buckling of rectangularplates under edge thrusts on the basis of IT. They usedenergy method to find the approximate value of crit-ical load. Pride and Heimerl2 showed a disagreementbetween the test results and the Handelman–Pragertheory1 which was based on IT; however, their resultswere well correlated with those obtained from DT.

Shrivastava3 showed that the decrease of plastic buck-ling stress parameters caused by transverse sheardeformation was more in IT than that of the DT.However, this difference in the reduction was notenough to compensate the discrepancy betweenthese two theories. Ore and Durban4 realized that incylindrical under uniaxial compression, the criticalloading obtained from both theories were quite simi-lar. Tugcu5 studied the effect of uniaxial load on plas-tic shear buckling coefficient. He concluded that theanalysis based on IT was more sensitive to interactiveloading and secant yield stress than DT. Durban6

realized that the IT predicted more buckling loadthan DT and the experimental data had more agree-ment with DT results. Durban and Zuckerman7

Proc IMechE Part G:

J Aerospace Engineering

2015, Vol. 229(7) 1280–1299

! IMechE 2014

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DOI: 10.1177/0954410014550047

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Department of Mechanical Engineering, Ferdowsi University of

Mashhad, Mashhad, Iran

Corresponding author:

M Kadkhodayan, Department of Mechanical Engineering, Ferdowsi

University of Mashhad, Mashhad 91775-1111, Iran.

Email: [email protected]

performed elastoplastic analysis for rectangular platesunder compression/tension using separation of vari-ables method; however, limited boundary conditionswere investigated. Smith et al.8,9 studied the plasticbuckling of steel plates under different loading condi-tions by using Rayleigh–Ritz method and based onCPT. Betten and Shin10 concluded that the bucklingof thin and thick plates could be in elastic and plasticranges, respectively. Mao and Lu11 studied the plasticbuckling of circular cylindrical shell subjected to axial,torsional, and circumferential loading. They con-cluded that DT gave more proper results comparedto IT which overestimated the critical loads; however,IT could predict better results for plastic torsionalbuckling. Wang et al.12 studied the elastoplastic buck-ling of thick plates using both IT and DT. They con-cluded that DT predicted less buckling stress factorand by increasing the thickness and the Ramberg–Osgood constants the differences between the resultsof two theories were increased. Kosel and Bremec13

solved the governing differential equation of elasto-plastic buckling of a circular annular plate by finitedifference method. They explained that rather thanelastic buckling, the elastoplastic results showed adecrease in buckling loads with increasing of plasti-city. El-Sawy et al.14 employed the finite elementmethod (FEM) to determine the elastoplastic buck-ling stress of uniaxially loaded square and rectangularplates with circular cutouts. They showed that thecritical buckling stress for perforated plates alwaysdecreased as the plate slenderness ratio increasedand that this reduction became steeper for largevalues of plate slenderness ratio, especially for smallhole sizes where the failure changed from elastoplasticinto pure elastic. Wang et al.15,16 showed that thebuckling coefficient decreased with increasing ofplate thickness and values of hardening as the platematerial approached the perfectly elastic–plastic con-stitutive relation. Moreover, the buckling stressesbased on DT were consistently lower than thoseof IT. Grognec and Levan17 studied the theoreticalaspects of elastoplastic buckling of plates and cylin-ders under uniform compression based on the 3Dplastic bifurcation theory assuming the IT with thevon Mises yield criterion and a linear isotropichardening. Wang et al.18,19 confirmed the reportedresults given by Durban and Zuckerman7 and Wanget al.12 Komur20 examined the effect of plate aspectratio, elliptical hole size, angle and location, and slen-derness ratio on elastoplastic buckling behavior. Hefound that as the plate slenderness ratio increased, thecritical buckling stress decreased for all perforatedplates. More recently, Robert et al.21 achieved a com-parison between the incremental deformation theoryand flow rule to simulate sheet-metal forming pro-cesses. A new algorithm based on incrementaldeformation theory (related to Hencky theory) waspresented, in order to compute the plasticity rule in

Abaqus. The major advantage of the new approachwas the time benefit when the material nonlinearitywas dominant. This formulation worked well whenthe strain path changed significantly under multiaxialstress state. More analyses and results in this subjectare mentioned in Chakrabarty.22 However, the pointthat the differences between the obtained results fromthe two theories of plasticity are considerable is still aparadox.

In this work, details of the governing equations ofplastic buckling of rectangular plates under variousloading and boundary conditions are introducedfirst. The first-order shear deformation and incremen-tal and deformation theories of plasticity areemployed. Thereafter, a GDQ method discretizationtechnique is used to solve these equations. Then, adirect iterative method is employed to obtain the elas-toplastic buckling coefficients. Convergence andaccuracy of the results are examined using availabledata in literature. The plastic behavior of plates isdescribed by the Ramberg–Osgood model. The effectsof thickness, aspect and loading ratios, material prop-erties, and different boundary conditions on the buck-ling coefficient are discussed. Contour plots ofbuckling mode shapes for various types of loading,aspect ratios, and boundary conditions are alsoillustrated.

Theoretical formulations

Figure 1 shows the geometry of rectangular platesunder in-plane uniform compressive and shear stres-ses where a, b, and h are the length, width, and thick-ness of the plate, respectively. The stress ratescorresponding to strain rate in plate are given by

_�x¼E � _"xþ� _"yþ� _�xy� �

, _�y¼E � _"xþ� _"yþ� _�xy� �

,

_�xy¼E � _"xþ� _"yþ _�xy� �

, _�xz¼2G _�xz, _�yz¼

2G _�yz

ð1Þ

Figure 1. The geometry of a rectangular plate under in-plane

compressive stresses.

Maarefdoust and Kadkhodayan 1281

where E is the Young’s modulus, G is the shear modu-lus (G is dependent on the type of analysis), and 2 isthe shear correction factor, and. Two theories of plas-ticity are used, i.e. the IT based on Prandtl–Reussequation and the DT based on Hencky equation.The incremental theory based on Prandtl–Reuss equa-tion was proposed by Prager (1938), Handelman et al.(1949), Hopkins (1949), Pearson (1950), and Besseling(1952).23 The constitutive equations of this theory canbe found in Wang et al.12 The parameters�, �, �, , �, �, and G are given by16

� ¼1

�c22c33 � c223� �

, � ¼1

�c13c23 � c12c33½ �,

� ¼1

�c11c33 � c213� �

, � ¼1

�c12c13 � c11c23½ �,

� ¼1

�c12c23 � c13c22½ �, ¼

1

�c11c22 � c212� �

,

� ¼E

T

c11 c12 c13

c21 c22 c23

c31 c32 c33

��, G ¼ E=2 1þ �ð Þ

ð2Þ

where

c11 ¼ 1� 3 1�T

E

� ��2y4�2eþ�2xy�2e

!,

c12 ¼�1

21�ð1� 2�Þ

T

E� 3 1�

T

E

� ��x�y2�2eþ�2xy�2e

!" #,

c13 ¼3

21�

T

E

� �2�x��y�e

� ��xy�e

� �,

c22 ¼ 1� 3 1�T

E

� ��2x4�2eþ�2xy�2e

!,

c23 ¼3

21�

T

E

� �2�y��x�e

� ��xy�e

� �,

c33 ¼ 2ð1þ�ÞT

E

� �þ 9 1�

T

E

� ��2xy�2e

!ð3Þ

where T and S are the tangent and secant moduli,respectively, which are calculated through stress–strain uniaxial curve. The deformation theory basedon Hencky equation was first discussed byKollbrunner (1946), Ilyshin (1947), Stowell (1948),Bijlaard (1949), and Alghazaly (1986).23–25 The par-ameters �, �, �, , �, and � are calculated by employ-ing equation (2) and G in this theory is given by

G ¼ E= 2þ 2�þ 3E

S� 1

� � ð4Þ

where

c11 ¼ 1� 3 1�T

S

� ��2y4�2eþ�2xy�2e

!,

c12 ¼�1

21�ð1� 2�Þ

T

E� 3 1�

T

S

� ��x�y2�2eþ�2xy�2e

!" #,

c13 ¼3

21�

T

S

� �2�x��y�e

� ��xy�e

� �,

c22 ¼ 1� 3 1�T

S

� ��2x4�2eþ�2xy�2e

!,

c23 ¼3

21�

T

S

� �2�y��x�e

� ��xy�e

� �,

c33 ¼ 3T

S�ð1� 2�Þ

T

E

� �þ 9 1�

T

S

� ��2xy�2e

!ð5Þ

Notice that by setting T¼S¼E, the parameters�, �, �, , �, and � are reduced to the parametersfor elastic buckling. Using strain energy, equation(1), and calculus of variations, the Euler–Lagrangedifferential equations associated with the minimiza-tion of the total potential energy functional, and thegoverning differential equations for elastoplasticbuckling of thick rectangular plates can be derived.

Solution methodology of thenonlinear equations

This method (GDQ) is practical and simple in sol-ving engineering problems. It was used in 1971 byBellman and Casti26 as a new technique for numer-ical solving of ordinary or partial equations. The firstwidespread use of this technique in the field of engin-eering problems was given by Bert and Malik.27 Thebenefit of accessing to a new and exact solution withthe least analyses in comparison to other numericalsolutions like finite element and boundary elementcauses the efficiency of this method to be revealedgradually. This method can solve higher order differ-ential equations with selecting few grid spacing. Itsother characteristics are simple application, program-ming, and high convergence rate.26,27 The GDQ dis-cretized form of the governing equations andboundary conditions at the domain and on theboundary grid points are presented. The naturaland simplest choice of the grid points is equallyspaced points in the direction of the coordinateaxes of the computational domain. It was demon-strated that nonuniform grid points gives a betterresults with the same number of equally spacedgrid points.27 Moreover, the distributions of gridspacing of Chebyshev–Gauss–Lobatto (C-G-L)

1282 Proc IMechE Part G: J Aerospace Engineering 229(7)

have the best convergence and highest accuracy28

which is used in the current study as following

xi ¼1

21� cos

i� 1

Nx � 1

� �, i ¼ 1, 2, . . . ,Nx,

yj ¼1

21� cos

j� 1

Ny � 1

� �, j ¼ 1, 2, . . . ,Ny

ð6Þ

Discretization of equations

The governing differential equations for elastoplasticbuckling of thick rectangular plates are written in dis-cretized form. To make Nx �Ny grid points on theplate where Nx and Ny represent the number of nodesin the x and y directions, respectively the GDQ dis-cretized form of the equations of elastoplastic buck-ling of rectangular plates at each grid point (i, j), withi ¼ 2, 3, . . . ,Nx � 1 and j ¼ 2, 3, . . . ,Ny � 1 become

2GhXNx

m¼1

Axim�

xmjþ

XNy

n¼1

Ayjn’

yinþXNx

m¼1

Bximwmjþ

XNy

n¼1

Byjnwin

!

¼�xhXNx

m¼1

Bximwmjþ�yh

XNy

n¼1

Byjnwin

þ2�xyhXNx

m¼1

XNy

n¼1

AximA

yjnwmn ð7Þ

�Eh3

12

XNx

m¼1

Bxim�

xmj þ

�Eh3

12

XNx

m¼1

XNy

n¼1

AximA

yjn�

ymn þ

�Eh3

12

�XNx

m¼1

XNy

n¼1

AximA

yjn�

xmn þ

XNx

m¼1

Bxim�

ymj

!þEh3

12

�XNy

n¼1

Byjn�

xin þ

XNx

m¼1

XNy

n¼1

AximA

yjn�

xmn

!

þ�Eh3

12

XNx

m¼1

XNy

n¼1

AximA

yjn�

xmn

þ�Eh3

12

XNy

n¼1

Byjn�

yin �

2Gh �xij þXNx

m¼1

Aximwmj

!¼ 0

ð8Þ

�Eh3

12

XNx

m¼1

Byjn�

yinþ

�Eh3

12

XNx

m¼1

XNy

n¼1

AximA

yjn�

xmnþ

�Eh3

12

�XNx

m¼1

XNy

n¼1

AximA

yjn�

ymnþ

XNy

n¼1

Bxjn�

xin

!þEh3

12

�XNx

m¼1

XNy

n¼1

AximA

yjn�

xmnþ

XNx

m¼1

Bxim�

ymj

!

þ�Eh3

12

XNx

m¼1

Bxim�

xmj

!þ�Eh3

12

�XNx

m¼1

XNy

n¼1

AximA

yjn�

ymn�

2Gh �yijþXNy

n¼1

Ayjnwin

!¼ 0

ð9Þ

where Axij,A

yij,B

xij,B

yij,wij, and �ij are the weighting

coefficients of the first- and second-order derivativeswith respect to the x and y, deflection, and rotation atgrid point ij, respectively. Application of GDQmethod to the governing equations leads to a set of3ðNx � 2Þ2 � 3ðNy � 2Þ2 equations with the samenumber of unknowns for all nodes of the domain.Finally, solving of generalized eigenvalue problemsystem of algebraic equations provides elastoplasticbuckling coefficient (the lowest eigenvalue) of theplate which can then be used to obtain the otherresultants using equations (7) to (9). The boundaryconditions are19

Simply supported edges (S)at x ¼ 0, x ¼ a

w1j ¼ wNxj ¼ 0, �y1j ¼ �yNxj¼ 0, �

XNx

m¼1

Axim�

xmj

þ �XNy

n¼1

Ayjn�

yin ¼ 0, j ¼ 1, . . . ,Ny

ð10Þ

and in y ¼ 0, y ¼ b

wi1 ¼ wiNy¼ 0, �xi1 ¼ �

xiNy¼ 0, �

XNx

m¼1

Axim�

xmj

þ �XNy

n¼1

Ayjn�

yin ¼ 0, i ¼ 1, . . . ,Nx

ð11Þ

Clamped edges (C)at x ¼ 0, x ¼ a

w1j ¼ wNxj ¼ 0, �x1j ¼ �xNxj¼ 0, �y1j ¼ �

yNxj¼ 0

ð12Þ


wi1 ¼ wiNy¼ 0, �xi1 ¼ �

xiNy¼ 0, �yi1 ¼ �

yiNy¼ 0

ð13Þ

Free edges (F)at x ¼ 0, x ¼ a

�XNx

m¼1

Axim�

xmj þ �

XNy

n¼1

Ayjn�

yin ¼ 0,

XNx

m¼1

Axim�

ymj þ

XNy

n¼1

Ayjn�

xin ¼ 0,

2Gh �xij þXNx

m¼1

Aximwmj

!¼ �xh

XNx

m¼1

Aximwmj ð14Þ



�XNx

m¼1

Axim�

xmj þ �

XNy

n¼1

Ayjn�

yin ¼ 0,

XNx

m¼1

Axim�

ymj þ

XNy

n¼1

Ayjn�

xin ¼ 0,

2Gh �yij þXNy

n¼1

Ayjnwin

!¼ �yh

XNy

n¼1

Ayjnwin ð15Þ

Now, the buckling coefficient K can be defined as

K ¼�chb

2

2Dð16Þ

where D ¼ Eh3=12ð1� �2Þ is the plate flexural rigidity.

Mechanical properties of material

The material used in this study is a typical aerospacealuminum alloy (AL 7075-T6). Here, the Ramberg–Osgood elastoplastic stress–strain relationship isused as

" ¼�

Eþk�0E

�

�0

� �c

ð17Þ

where " is the total plastic strain and c and k arematerial parameters. The tangent and secant moduliused in the equation are calculated as follows

E

T¼ 1þ ck

�

�0

� �c�1

,E

S¼ 1þ k

�

�0

� �c�1

, ðc4 1Þ

ð18Þ

The characteristics of the metal are obtained bymeans of equation (17) and are shown in Table 1.The shear correction factor 2 ¼ 5=6 has been usedin all calculations.

Verification of analysis

To show the efficiency and accuracy of the presentnumerical method, the results of the analysis are com-pared with other available data. It is assumed that the

stress is applied uniformly as �x ¼ ��P, �y ¼ P, and�xy ¼ 0 where � is the loading ratio. As it is shown inFigure 1, the loading ratios � ¼ 1 for biaxial compres-sion/tension, � ¼ 0 for axial compression and � ¼ �1for equibiaxial compression are considered. TheRamberg–Osgood stress–strain relation for materialAL7075-T6 for different values of c described byequation (17) is depicted in Figure 2. To increasethe accuracy of the analysis, the grid points have tobe selected properly. In the present analysis, toachieve the best convergence rate, it takes thatNx¼Ny¼N and the C-G-L grid spacing should beused (equation (6)). Taking a 13 grid points gives anexcellent convergence rate for GDQ method(Figure 3). The type of loading and the value offactor c affects the convergence rate significantly. Acomparison between the obtained results from DTand experimental data for rectangular plates underuniaxial compression are presented in Figure 4. Itcan be seen that the results attained by DT are closeenough to the experimental ones. To validate the pre-sented results, the buckling coefficient subjected touniaxial and equibiaxial loading with SSSS squareplates are compared with the published experimentaland numerical data and then more new results arepresented (Table 2). It is seen that the results are invery good agreements with other reported data.

Numerical results and discussion

In the following, three different loading namely uni-axial and biaxial, pure shear and combined shear, anduniaxial loading are considered and the obtainedresults are discussed in detail. The parametric effectof aspect, thickness and loading ratios, material prop-erties, incremental and deformation theories, bound-ary condition, and shear correction factor on theelastoplastic buckling coefficient are studied andsome mode shapes of thick plates are illustrated.

Uniaxial and biaxial loading

Figure 5 shows the variation of the aspect ratio versusbuckling coefficient in uniaxial compression for

Figure 2. Ramberg–Osgood stress–strain relation.

Table 1. Material properties and Ramberg–Osgood

parameters.29,30

Material

Properties

E=�0 c k �

AL 7075-T6 150 9.2 3/7 0.33

AL 2014-T6 204.58 20 3/7 0.33


various boundary conditions using IT and DT alongwith different Ramberg–Osgood constant c. As it isseen, with the decrease in the strain hardening(increasing of c) the buckling coefficients are reduced.For the similar conditions, the buckling coefficientsobtained from DT are less than those of the IT.Figure 5(d) shows that in the case of FCSC with uni-axial compression loading, increasing of aspect ratioreduces the buckling coefficient. When the aspect ratioincreases in FCFC plates, the buckling coefficient firstshows an increase, then a decrease and after that nosignificant change is observed. Moreover, for thehigher values of Ramberg–Osgood constant (elastic/perfectly plastic properties) less variations areobserved.

In SCSC and CCCC plates under equibiaxial com-pression loading, with increasing the aspect ratio, thebuckling coefficient decreases monotonically (Figure6(a) and (b)). Moreover, using higher values ofRamberg–Osgood constant causes the resultsobtained from IT and DT become closer to eachother. In the FCFC and FCSC plates, unlike theother cases, the buckling coefficients for Ramberg–Osgood constant of c ¼ 20, obtained from deform-ation theory are more than those of c¼ 2 and 3(Figure 6(c) and (d)). Hence, in the plates with freeboundary conditions the buckling coefficients increasewhen material properties become closer to the elastic-perfectly plastic case. With increasing the aspect ratio,the buckling coefficients have more rapid variations inthe range of 0:54a=b42 rather than after 2 (Figures 5and 6). However, for the FCSC case, after an initialincreasing of buckling coefficients a gradual decreas-ing happens. This rate is reduced by raising ofRamberg–Osgood constant c (Figure 6(d)).

Figures 7 and 8 show that decreasing of yieldingstrain (increasing of E=�0) for both uniaxial and equi-biaxial compression loading reduces the bucklingcoefficient. The decreasing rate is higher when DT isused and it rises by using larger values of c andclamped edge boundary conditions (Figures 7(b) and(8b)). Adding more free edges boundary conditionsand increasing the Ramberg–Osgood constant inboth uniaxial and equibiaxial compression loadingcan increase the closeness of the results obtainedfrom two theories of plasticity (Figures 7(c, d) and8(c, d)). It is found that by decreasing the strainhardening in biaxial compression loading, the buck-ling coefficients obtained from two theories becomequite close to each other, which are more obvious inequibiaxial rather than uniaxial loading.

Generally, raising the thickness ratio decreases thebuckling coefficient for both IT and DT and different

Figure 3. Convergence of the results for different boundary conditions.

IT: incremental theory; DT: deformation theory.

Figure 4. A comparison between the buckling stresses

obtained by IT and DTwith numerical and experimental results

for SSSS square plates.

IT: incremental theory; DT: deformation theory; FEM: finite

element method; GDQM: generalized differential quadrature

method.


boundary conditions (Figures 9 and 10). In thin platesðh=b5 0:05Þ, however, the buckling coefficientsdecrease more rapidly for larger values of c whichare more significant when the DT is used. In addition,the results obtained by IT and DT are closer to eachother in equibiaxial compression rather than uniaxialone. It is seen that for the thicker plates,ð0:054h=b40:1Þ, the effect of thickness on the

buckling coefficient become less for the higher valuesof c, which is more apparent for equibiaxial compres-sion. Generally, it is seen that the DT predicts lowerbuckling coefficients compared to IT. Moreover,increasing the thickness, reducing the strain harden-ing, and having more clamped boundaries lead tomore differences between the results obtained fromtwo theories. Moreover, adding more free edges to

(a) (b)

(d)(c)

Figure 5. Variations of buckling coefficient versus aspect ratio for rectangular plates subjected to uniaxial in-plane compressive loads

with various boundary conditions ðh=b ¼ 0:025Þ.


Table 2. Comparison of buckling coefficient for SSSS square plates under uniaxial and equibiaxial loading.

Loading Source

a/h¼ 20 a/h¼ 10

IT DT IT DT

Uniaxial Handleman and Prager1 3.5740 – 3.5193 –

Shrivastava3 3.5278 2.8058 3.4636 0.9205

Wang et al.12 3.4955 2.7954 3.1908 0.9144

Wang and Aung16 3.4955 2.7954 3.1908 0.9144

Present study 3.4955 2.7954 3.1908 0.9144

Equibiaxial Durban and Zuckerman7 1.8713 1.8649 0.6661 0.7346

Wang et al.12 1.8713 1.8649 0.8084 0.7300

Wang and Aung16 1.8713 1.8649 0.8084 0.7300

Present study 1.8713 1.8649 0.8084 0.7300



(a) (b)

(d)(c)

Figure 6. Variations of buckling coefficient versus aspect ratio for rectangular plates subjected to equibiaxial in-plane compressive

loads with various boundary conditions ðh=b ¼ 0:025Þ.


(a) (b)

(d)(c)

Figure 7. Variations of buckling coefficient versus E=�0 for square plates subjected to uniaxial in-plane compressive loads with

various boundary conditions ðh=b ¼ 0:025Þ.



(a) (b)

(d)(c)

Figure 8. Variations of buckling coefficient versus E=�0 for rectangular plates subjected to equibiaxial in-plane compressive loads

with various boundary conditions ðh=b ¼ 0:025Þ.


(a) (b)

(d)(c)

Figure 9. Variations of buckling coefficient versus thickness ratio for square plates subjected to uniaxial in-plane compressive loads

with various boundary conditions.



the plate make those differences smaller. It can befound from Figures 9 and 10 that for the plate thick-ness of h=b40:02, increasing the parameter c raisesthe buckling coefficient when DT is used. Theranges of thicknesses for different boundary condi-tions are listed in Table 3. In thicker plates,ðh=b50:05Þ, however, increasing of Ramberg–

Osgood constant ðc ¼ 20Þ has no significant effect onthe bucking coefficients for plates subjected to equi-biaxial compression loading when IT is used, but itseffect for DT is visible (Figure 10).

Figure 1 shows that the load applied in the axialdirection as �x ¼ ��P and changing the loading ratio,�, will affect the buckling coefficient. Figure 11 showsthe comparison of IT and DT results for SSSS squareplates for different thicknesses and loading ratios. Forthe case of h=b ¼ 0:05, there is a close agreementbetween two theories of plasticity in the region of�14�4� 0:6. However, increasing of loading andthickness ratios lead to increasing of differencesbetween two theories (Figure 11(c)). If during calcu-lations, the predicted stresses become more than theyielding stress, those results are unacceptable. Theacceptable and unacceptable results are shown bysolid and dash lines, respectively, which happens forbiaxial compression/tension loading (Figure 11).There is no agreement between the obtained bucklingcoefficients for thickness ratios of h=b ¼ 0:1 andh=b ¼ 0:2 from two theories (Figure 11(b) and (c)).Moreover, no acceptable results can be obtained byusing IT for h=b ¼ 0:2. Generally, when the appliedload is biaxial compression/tension, �4 0, increasingof plate thickness leads to rising of differences

(a) (b)

(d)(c)

Figure 10. Variations of buckling coefficient versus thickness ratio for square plates subjected to equibiaxial in-plane compressive

loads with various boundary conditions.


Table 3. The range of thickness ratio in DT that buckling

coefficient increases with increasing Ramberg–Osgood

constant.

B.C

Loading

Uniaxial Equibiaxial

SSSS h=b40:021 h=b40:026

SCSC h=b40:015 h=b40:02

SSCC h=b40:015 h=b40:02

SCCC h=b40:013 h=b40:018

SSSC h=b40:018 h=b40:025

CCCC h=b40:013 h=b40:015

FCCC h=b40:021 h=b40:023

FCFC h=b40:022 h=b40:024

FCSC h=b40:021 h=b40:025


between the two theories and the results obtainedfrom IT become less reliable. Furthermore, when ITis used, with increasing the thickness and loading ratioin �4 0, the tangent modulus T becomes almost zero

and the obtained results are not reliable anymore(Figure 12(a)). Development of plastic zone leads toreduction of plate stiffness and the buckling loadsbecome smaller.

(a) (b)

(c)

Figure 11. Comparison of IT and DT predictions of buckling coefficient for SSSS square plates under various loading and thickness

ratios.


(a) (b)

Figure 12. Variations of tangent modulus versus loading ratio for SSSS square plates with various thickness ratios: (a) IT; (b) DT.



When the loading and thickness ratios increase andthe plastic zones develop, DT predicts smaller valuesfor shear modulus and, in turn, the buckling loads(Figures 11 and 13). In fact, different moduli areused by two theories and the effect of thickness ratiois more significant than that of loading ratio in plasticzone development. Figure 14 shows the effect of shearcorrection factor on the buckling coefficient. In thinplates of h=b ¼ 0:001, the buckling coefficientsachieved from thin and thick plate theories, for bothbiaxial compression and biaxial compression/tensionloading, are similar and the shear correction factorhas no noticeable effect on the results (Figure 14(a)).Moreover, there is a good agreement between theresults obtained from IT and DT theories with orwithout transverse shear deformation effect. Theeffect of shear correction factor increases as the thick-ness of plate increases. However, the CPT predictslarger values for buckling coefficients in comparisonwith the thick plate theory because the former neglectsthe effect of transverse shearing deformation (Figure14(b) and (c)). In the biaxial compression/tensionloading ð05 �41:5Þ, the most differences areobserved between IT and DT and also between thin

and thick plate theories. Furthermore, in the case ofh=b ¼ 0:2, there is no agreement between the resultsachieved from IT and DT and also between the thinand thick plate theories (Figure 14(c)).

Reissner and Mindlin proposed that 2 ¼ 5=6 and2 ¼ 2=12, for shear correction factor, respectively.Table 4 indicates the effect of shear correction factoron the buckling coefficient under equibiaxial loadingin various thickness ratios for SSSS thin and thickplates. An increase of shear correction factorð2 ¼ 1000Þ makes the Reissner–Mindlin theorycloser to the CPT. As it is observable from Table 4,the effect of shear correction factor becomes moresignificant when the thickness of plate increases,which reduces the buckling coefficients. Moreover,for the thicker plates the influence of shear correctionfactor is more in IT than that in DT.

Pure shear loading

Table 5 shows a good agreement between the bucklingcoefficient for SSSS and CCCC square plates underpure shear loading for IT and DT and other reporteddata. Figure 15 displays the shear buckling coefficient,

(a) (b)

(c)

Figure 13. Shear modulus versus loading ratio for IT and DT for SSSS square plates with various thickness ratios.



K ¼ �xyhb2=ð 2DÞ, versus the aspect ratio for pure

shear loading in different boundary conditions.Increasing of c, similar to uniaxial and equibiaxialloading, decreases the shear buckling coefficient inboth IT and DT analyses. Moreover, the bucklingloads attained from DT are smaller than thoseof IT. The buckling coefficients in pure shear loadingare larger than those of axial and equibiaxial com-pression loading (Figures 5, 6, and 15).

The most shear buckling coefficient variationsoccur in the range of a=b5 1:5 and increasing ofaspect ratio reduces the variation of buckling coeffi-cient. It is seen that adding more free boundary con-ditions in pure shear loading can affect the variationon buckling coefficient significantly. Comparison ofFigure 15(c) and (d) shows that for the case ofFCFC, increasing the aspect ratio causes a kind offluctuation on the values of shear buckling coefficient;

(a) (b)

(c)

Figure 14. Comparison of load ratios versus buckling coefficient for square plates with and without transverse shear deformation by

IT and DT with various thickness ratios.


Table 5. Comparison of shear buckling coefficient for square

plates under pure shear loading with SSSS and CCCC boundary

conditions h=b40:025.

B.C Source

a/b¼ 1 a/b¼ 4

IT DT IT DT

SSSS Gerard32,33 – 5.3610 – 4.6187

Becker and Gerard34 – 6.4507 – 5.1330

Wang and Aung16 6.7814 6.2008 5.1325 5.0377

Present study 6.7812 6.1992 5.1320 5.0370

CCCC Gerard32,33 – 5.8463 – 5.3573

Becker and Gerard34 – 7.2690 – 6.4440

Wang and Aung16 11.1769 7.1889 7.4231 6.3658

Present study 11.1767 7.1879 7.4225 6.3642


Table 4. Effects of shear correction factor 2 on the buckling

coefficient K under equibiaxial loading in various thickness

ratios for SSSS square plates.

h/b

Reissner–Mindlin

theory 2 ¼ 5=6Reissner–Mindlin

theory 2 ¼ 1000

Classical plate

theory

IT DT IT DT IT DT

K 0.001 2.0000 2.0000 2.0000 2.0000 2.0000 2.0000

0.05 1.8713 1.8649 1.8896 1.8830 1.8896 1.8830

0.075 0.9127 1.1476 0.9257 1.1542 0.9257 1.1542

0.1 0.8083 0.7300 0.8168 0.7346 0.8168 0.7346

0.2 0.6209 0.2237 0.6700 0.2270 0.6700 0.2270



however, it is reduced with rising of c when DT isused. In pure shear loading, the results obtainedfrom IT and DT have larger discrepancy comparedto uniaxial and equibiaxial loading. This discrepancyis reduced in axial compression loading and decreasesagain in equibiaxial compression.

Figure 16 shows that increase in E=�0, decreasesthe shear buckling coefficient. Moreover, the discrep-ancy between IT and DT increases with increase inE=�0 and c. It is also seen that the rate of decreasingof buckling coefficient with increasing of E=�0 in shearloading is larger than those of both uniaxial and equi-biaxial loading (Figures 16(b–c), 6(b–c), and 8(b–c)).Figure 17 shows the effect of transverse shear deform-ation on the shear buckling coefficient. As it is seen,when the thickness ratio increases the shear bucklingcoefficient decreases. Moreover, by increasing thethickness ratio, the discrepancy of shear bucklingcoefficient of IT and DT results increases which ismore severe for higher values of c. It is also noticedthat increasing of c affected the DT results moreseverely rather than IT ones (Figure 17(a) to (d)).The effect of transverse shear deformation on theplastic buckling coefficient is more visible than thatof elastic buckling. Moreover, with decreasing thestrain hardening or yielding strain the variation of

buckling coefficients are small in IT which is not truein DT. Therefore, DT is more sensitive to materialproperties (E=�0 and c) rather than IT. In otherwords, DT predicts more accurate results than IT forelastic-perfectly plastic materials. The variation ofbuckling mode shapes for uniaxial and equibiaxialcompression and pure shear loading for differentboundary conditions and aspect ratios are shown inFigure 18. As it is seen, the buckling occurs in highermode shapes as the aspect ratio increases. The reduc-tion of buckling coefficient occursmore slightly in equi-biaxial compression and for some cases no change inmode shape is visible (Figures 5, 6, and 18). Moreover,it is difficult to distinguish the location of kinks inFigure 15(a) but the mode shape shift is apparent inthis case. It should also be mentioned that the kinkslocations depend on the plasticity theories used.

Combined shear and uniaxial loading

For plates subjected to two type of loading simultan-eously, the critical combinations are usually given bymeans of interaction curves, which can be used toobtain the value of one load required to producebuckling when the value of another load is known.If it is assumed that �y ¼ 0 in Figure 1, then a

(a) (b)

(d)(c)

Figure 15. Variations of shear buckling coefficient versus aspect ratio for rectangular plates under shear stress with various

boundary conditions ðh=b ¼ 0:025Þ.



(a) (b)

(d)(c)

Figure 17. Variations of shear buckling coefficient versus thickness ratio for square plates under shear stress with different

boundary conditions.


(a) (b)

(d)(c)

Figure 16. Variations of shear buckling coefficient versus E=�0 for square plates under shear stress with various boundary condi-

tions ðh=b ¼ 0:025Þ.



combined shear and uniaxial loading is applied. Whenthe loading is a combined shear and axial compres-sion, the Stein and Batdorf equation for inelasticbuckling interaction curve is35

�x�xÞcr

� �þ

�xy�xyÞcr

� �2

¼ 1 ð19Þ

This equation estimates a lower bound values forbuckling coefficient. The equation was also defined byPeters36 for plastic state. It estimates an upper boundvalues for buckling coefficient. Now suppose that

Rx ¼�x�xÞcr

Rxy ¼�xy�xyÞcr

ð20Þ

where Rx is the ratio of uniaxial stress to critical uni-axial stress and Rxy is the ratio of shear stress to crit-ical shear stress due to pure shear. If Kx and Kxy are

the axial and shear buckling coefficients respectively,then Kxy coefficient can be calculated from knownvalues of Kx. As it is observable from Table 6, thereis a good agreement between the results of presentstudy and those of Batdorf and Stein. Figure 19shows the interaction curve of buckling coefficientfor SSSS and CCCC rectangular plates under

Figure 18. Buckling mode shapes of SSSS rectangular plates under pure shear in various aspect ratios ðh=b ¼ 0:025Þ.

Table 6. Comparison of shear buckling coefficient Kxy for

SSSS plates under combined loading.

Source

a/b

1 2 4

Kx �2 1 3 �2 1 3 �4 1 3

Kxy Batdorf

and Stein35

11.56 8.11 4.68 – 5.71 3.29 7.95 5 2.94

Present

study

11.49 8.06 4.66 7.89 5.67 3.28 7.93 4.90 2.93


combined shear and uniaxial compression (or tension)stresses and different aspect ratios with DT. It is seenthat for combined shear and uniaxial compressionloading, increasing of axial buckling coefficient ðKxÞ,causes decreasing of shear buckling coefficient ðKxyÞ

for different aspect ratios. On the other hand, in com-bined shear and uniaxial compression loading, theplate buckles in lower shear stress when more com-pression stress is applied. The decreasing rate isreduced with increasing of aspect ratio. However, incombined shear and uniaxial tension loading, byincreasing the uniaxial buckling coefficient ðKxÞ, theshear buckling coefficient ðKxyÞ increases. Moreover,

the plate buckles in higher shear stress when moretensile stress is applied. Moreover, Figure 19(a) and(b) shows that in CCCC plates under uniaxial com-pression stress, the buckling occurs at larger shearstress compared with SSSS plates. As a result,adding more clamped boundary conditions not onlyincrease axial and shear buckling coefficients but alsoincrease their variations. According to Figure 20, inthe case of pure shear and axial tensile, principle ten-sile stress is uniformly formed at 45� and is distributedacross the plate. Whereas in the case of axial tensile,the plate has uniform stresses. Therefore, the additionof tensile stresses increases the buckling effect of

Figure 20. States of combined stresses.

(a) (b)

Figure 19. Interaction curve for combined shear and axial loading for SSSS and CCCC rectangular plates for various aspect ratios in

terms of buckling coefficients kx and kxy ,ðh=b ¼ 0:001, c ¼ 5Þ.


pure shear stresses and, on the other hand, the com-pressive stress reduces the buckling effect of pureshear stresses.

Figure 21 shows that the interaction curve undercombined shear and axial loading obtained by DT forvarious thickness ratios are enclosed by their elastic35

and plastic buckling36 curves. For instance, there is anelastic buckling in thickness ratio of h=b ¼ 0:0001which is consistent with Stein’s curve. But, when thethickness ratio increases the results become closer toPeters’s curve. Buckling mode shapes related to thepure shear and uniaxial loading are also shown inFigure 21.

Conclusions

In this paper, the elastoplastic buckling of rectangularplates were investigated using two theories of plasti-city, i.e. IT and DT. The generalized differential quad-rature method as an accurate, simple, andcomputationally efficient numerical tool was adoptedto discretize the governing equations and the relatedboundary conditions. The results were verified byothers available data and several interesting newresults were found. The effects of aspect, thicknessand loading ratios, material properties, shear correc-tion factor, and different boundary conditions wereinvestigated to evaluate the buckling load in uniaxial,equibiaxial, pure shear and combined shear, and uni-axial compression/tension loading. The interactioncurve of buckling coefficient under combined shearand uniaxial loading for various aspect and thicknessratios were showed. Finally, the shapes of plasticmode were given for various loading and differentboundary conditions and aspect ratios. Some generalinferences are mentioned as below:

. The results were shown that when the plasticzone develops, the deformation theory predicts

lower buckling loads compared to incrementaltheory.

. The agreement between the results of IT and DTincreases with adding more free edge boundaryconditions. Furthermore, by increasing of loadingratio in biaxial compression/tension, the differencesbecome larger.

. The effect of shear correction factor is more in ITrather than DT.

. In pure shear loading, the results obtained from ITand DT have larger discrepancy compared to uni-axial and equibiaxial loading. This discrepancydecreases in axial compression loading anddecreases again in equibiaxial compression one.

. DT is more sensitive in material properties ratherthan IT. In other words, DT predicts better resultsthan IT for elastic-perfectly plastic materials.

. For combined shear and uniaxial compressionloading, increasing the axial buckling coefficientreduces the shear buckling coefficient. The decreas-ing rate is reduced with increasing of aspect ratio.

Funding

This research received no specific grant from any fundingagency in the public, commercial, or not-for-profit sectors.

Conflict of interest

None declared.

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Appendix

Notation

a length of platesAx

ij,Ayij weighting coefficients of the first-

order derivative in x and y direc-

tions, respectivelyb width of platesBxij,B

yij weighting coefficients of the

second-order derivative in x and y

directions, respectivelyc, k Ramberg–Osgood parametersD flexural rigidity of platesE Young’s modulus of elasticityG shear modulus of elasticityh thickness of platesh=b thickness ratioK buckling coefficient


Mxx,Myy,Mxy bending moments about y- and x-axes and twisting moment,respectively

Nx,Ny number of grid points in x and ydirections

Rx ratio of uniaxial stress to criticaluniaxial stress

Rxy ratio of shear stress to criticalshear stress due to pure shear

Sij stress deviator tensorSðEsÞ secant modulusTðEtÞ tangent modulusu, v,w displacement components

�,�, �,�,�, parameters used in stress–strainrelations

" total plastic strain"e total effective strain�xy, �xz, �yz shear strain components2 shear correction factor�x,�y rotations about x, y�x, �y normal stress�e effective stress�xy, �xz, �yz shear stress components� Poisson’s ratio� loading ratio


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